A simple example...
exists...
doesn't exist...
Kind regards
I'm trying, unsuccessfully at the moment, to show the following, which I know is true: If is Riemann integrable, then is Riemann integrable. And, I need to do this without the u,v integrable implies uv integrable. I know there's a way to do it, so if you can point me in the right direction, that would be great! Thanks!
Okay, this may be a slight difference of definition, but here is the issue:
A function is is Riemann Integrable on if there is an such that for all there exists such that for every tagged partition of with , then
Now, if you look at the Riemann sum part you have
but if I choose my tag of the first sub-interval to be 0, ie. then which is undefined. So, and therefore we cannot apply the FTC.
simply is not a function from into because 0 is not in the domain.
What is your definition?
Exactly, By definition, 1/sqrt{x} is not Riemann integrable, so the theorem that show f-->f^2 does not apply. Hence this does not break the rules. After all, a limit of an integral is a limit of a limiting process, which is where everything gets alittle funny. So, in conclusion, we are both right: f integrable implies f^2 integrable, but improper integrals do not necessarily obey the same rules.
has confused problems from integration theory with an improper integral from basic calculus.
In basic calculus, exist and equals .
But it is also known as an improper integral because is not defined at .
In integration theory the function is not Riemann integrable on , not being bounded there.
But it does have an improper integral there.
The original Riemann definition is reported here...
http://en.wikipedia.org/wiki/Riemann_integral
... and here we can read...
Given an f(*) defined in the closed interval , the finite sequence , the finite sequence subject to the condition , a real number , if exists a real number so that for any partition is...
(1)
... we define...
(2)
Now I have indicated as example the function...
, (3)
It is obvious that the (3) is not defined in , but, as in most of such cases, we can resolve the abiguity defining f(*) [for example...] as follows...
for , for (4)
... so that is...
for , for (5)
Now we have to verify if defined in (4) is Riemann integrable according to (1) and if defined in (5) is not Riemann integrable according to (1)...
Kind regards
Now we try to compute the 'Riemann sum' ...
(1)
... having as goal to arrive to the integral...
(2)
The most simple choice is...
for , for with (3)
Defining...
in (4)
... the integral (2) is...
(5)
The limit (5) has been computed in elegant fashion in ...
http://www.mathhelpforum.com/math-he...-sequence.html
Kind regards
Obviously if I want to compute the integral of 1/sqrt{x} I can, it can be done is second quarter calculus. The point was that 1/sqrt{x} is NOT Reimann integrable in the strictest, by definition, sense. As a counterexample to f integrable --> f^2 integrable you MUST start with a function that is Reimann integrable. 1/sqrt{x} simply is not on [0,1]. It doesn't matter how you alter the function so that it becomes defined, for it is obviously then not the same function. So the implication is ambiguous.